Finite Group Databasebeta

← #27152217

Subgroups — #27152217

Order 12416.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690836C212yesmaximal, complemented
2C2×#15690832C211yesmaximal, characteristic, complemented
3#20261302C212yesmaximal, complemented
4#20261123C214yesmaximal, complemented
5#20261118C212yesmaximal, complemented
6#20261110C212yesmaximal, complemented
7#20260690C211yesmaximal, characteristic, complemented
8C22×D8×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2212yes—
10#15690836—28no—
11#15690832—24no—
12C2×Dic388C2211yescharacteristic, complemented
13C2×#13095500C2212yescomplemented
14C2×#13095499C2214yescomplemented
15C2×D776C2211yescharacteristic, complemented
16C2×#13095496C2214yescomplemented
17C2×#13095495C2211yescharacteristic
18#15690806C2214yescomplemented
19#15690797C2214yescomplemented
20#15690781—24no—
21#15690779—24no—
22#15690769C2211yescharacteristic
23#15690720C2212yescomplemented
24C8×Dic97C2212yescomplemented
25C22×D4×C97C2212yes—
26C2×D8×C97—24no—
27C2×D8×C97C2212yes—
28C2×D8×C97C2212yes—
29C22×C8×C97C2211yescharacteristic
30#13095511—24no—
31#13095511—24no—
32C2×Dic194C2312yes—
33C2×D388C2312yes—
34C2×C22⋊D97—22no—
35C2×C22⋊D97—28no—
36C2×C4×D97—22no—
37Dic388—22no—
38#13095500—24no—
39#13095499—28no—
40D776—22no—
41#13095496—28no—
42#13095495D414yescomplemented
43C2×#11715895C2312yes—
44C4⋊Dic97D412yes—
45C22⋊Dic97—28no—
46C22⋊Dic97D412yes—
47C4×Dic97C2312yes—
48C24×C97—22no—
49C2×D4×C97—24no—
50C2×D4×C97—24no—
51C2×D4×C97—22no—
52C2×D4×C97C2314yes—
53C22×C4×C97C2311yescharacteristic
54D8×C97—24no—
55D8×C97—24no—
56C2×C8×C97D414yescomplemented
57C2×C8×C97C2311yescharacteristic
58C2×C8×C97C2311yescharacteristic
59C23×C97—24no—
60C23×C97—44no—
61C23×C97—84no—
62C23×C97C2×D411yescharacteristic
63C22×D97—22no—
64D4×C97—24no—
65D4×C97—24no—
66D4×C97—44no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D412yes—
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2411yescharacteristic
71C22⋊D97—44no—
72C22⋊D97—88no—
73C2×Dic97—22no—
74C2×Dic97C2×D414yes—
75D388—22no—
76D388—22no—
77C4×D97—44no—
78Dic194—22no—
79Dic194—22no—
80C8×C97C2×D412yes—
81C8×C97C2×D412yes—
82#11715895C2×D414yes—
83C22×C97—22no—
84C22×C97—44no—
85C22×C97—48no—
86C22×C97—88no—
87C22×C97#12712yes—
88C22×C97C22×D411yescharacteristic
89C2×D97—42no—
90C2×D97—44no—
91C4×C97#12712yes—
92C4×C97C22×D411yescharacteristic
93C4×C97C22×D411yescharacteristic
94Dic97—24no—
95Dic97—42no—
96C2×C97—22no—
97C2×C97—84no—
98C2×C97C2×#12711yescharacteristic
99C2×C97#57612yes—
100D97—82no—
101#3155—971nomaximal, Sylow
102C97#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—972no—
104C2×#135—971no—
105C22×D8D9711yescharacteristic, complemented
106#535—972no—
107#496—974no—
108#495—972no—
109#493—972no—
110#405—971no—
111C2×#41—972no—
112C22×D4C2×D9712yes—
113AGL(1,ℤ/8)—1948no—
114#135—1944no—
115C2×Q16—971no—
116C2×SD16—974no—
117C2×SD16—972no—
118C2×D8—24no—
119C2×D8—974no—
120C2×D8—971no—
121C2×D8C2×D9712yes—
122C2×D8C2×D9712yes—
123C2×M16—971no—
124C22×C8C2×D9711yescharacteristic
125#127—974no—
126#124—974no—
127#121—1944no—
128#120—1944no—
129#117—971no—
130#97—972no—
131C4×C8—972no—
132C24—22no—
133#41—1944no—
134#41—1944no—
135C2×Q8—972no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—972no—
140C2×D4—1948no—
141C2×D4—1942no—
142C2×D4C22×D9714yes—
143C22×C4—1942no—
144C22×C4C22×D9711yescharacteristic
145Q16—1942no—
146SD16—1948no—
147SD16—1944no—
148D8—24no—
149D8—24no—
150D8—1948no—
151D8—1942no—
152M16—974no—
153C2×C8—972no—
154C2×C8C22⋊D9714yescomplemented
155C2×C8C22×D9711yescharacteristic
156C2×C8C22×D9711yescharacteristic
157C4⋊C4—972no—
158C22⋊C4—972no—
159C22⋊C4—1948no—
160C42—972no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—1942no—
165C23D4×D9711yescharacteristic
166Q8—1942no—
167Q8—1942no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—7768no—
172D4—3884no—
173D4—1942no—
174D4—1942no—
175C2×C4—974no—
176C2×C4—3884no—
177C2×C4—1942no—
178C2×C4D4×D9711yescharacteristic
179C2×C4D4×D9712yes—
180C2×C4C2×C22⋊D9712yes—
181C2×C4C23×D9711yescharacteristic
182C8—974no—
183C8C2×C22⋊D9712yes—
184C8C2×C22⋊D9712yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—3882no—
190C22—3884no—
191C22#1569080612yes—
192C22C2×D4×D9711yescharacteristic
193C4—3882no—
194C4—1944no—
195C4#1569080612yes—
196C4C2×D4×D9711yescharacteristic
197C4C22×C22⋊D9711yescharacteristic
198C2—22no—
199C2—84no—
200C2—7762no—
201C2C2×#1569080611yescharacteristic
202C2#2026153612yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×D8×C97C2
socleC22×C97C22×D4
hypercenterC22×D8D97
hypocenterC97#3155
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C2×#15690832C2semidirect
3. #20261302C2semidirect
4. #20261123C2semidirect
5. #20261118C2semidirect
6. #20261110C2semidirect
7. #20260690C2semidirect
8. C22×D8×C97C2semidirect
12. C2×Dic388C22semidirect
13. C2×#13095500C22semidirect
14. C2×#13095499C22semidirect
15. C2×D776C22semidirect
16. C2×#13095496C22semidirect
18. #15690806C22semidirect
19. #15690797C22semidirect
23. #15690720C22semidirect
24. C8×Dic97C22semidirect
42. #13095495D4semidirect
56. C2×C8×C97D4semidirect
102. C97#3155semidirect
105. C22×D8D97semidirect
154. C2×C8C22⋊D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156908362
C2×#156908322
#202613022
#202611232
#202611182
#202611102
#202606902
C22×D8×C972
#315597